On a conjecture of Ashbaugh and Benguria about lower eigenvalues of the Neumann Laplacian (Q2697579)

Let \(0=\mu_0(\Omega)<\mu_1(\Omega)\leq\cdots\leq\mu_j(\Omega)\leq\cdots\nearrow+\infty\) denote the Neumann eigenvalues of the Laplacian on \(\Omega\), a bounded smooth domain in a complete \(n\)-dimensional Riemannian manifold \(M\). When \(M=\mathbb R^n\) the well-known Szegő-Weinberger inequality states that \[ |\Omega|^{2/n}\mu_1(\Omega)\leq\omega_n^{2/n}\mu_1(B(0,1)), \] where \(|\Omega|\) is the Lebesgue measure of \(\Omega\) and \(\omega_n\) is the measure of the unit ball \(B(0,1)\) in \(\mathbb R^n\). Equality holds if and only if \(\Omega\) is a ball. A similar results holds when \(M=\mathbb H^n\) is the standard hyperbolic space of curvature \(-1\) (Ashbaugh-Benguria): \[ \mu_1(\Omega)\leq\mu_1(B) \] where \(B\) is a geodesic ball with \(|\Omega|=|B|\), with equality holding if and only if \(\Omega\) is a geodesic ball. The authors improve the previous inequalities in the following sense. \begin{itemize} \item[1.] If \(M=\mathbb R^n\) then \[ \sum_{j=1}^{n-1}\frac{1}{|\Omega|^{2/n}\mu_j(\Omega)}\geq\frac{n-1}{\omega_n^{2/n}\mu_1(B(0,1))}, \] with equality if and only if \(\Omega\) is a ball. \item[2.] If \(M=\mathbb H^n\) then \[ \sum_{j=1}^{n-1}\frac{1}{\mu_j(\Omega)}\geq\frac{n-1}{\mu_1(B)} \] where \(B\) is a geodesic ball with \(|\Omega|=|B|\), with equality if and only if \(\Omega\) is a geodesic ball. \end{itemize} This result supports a conjecture of Ashbaugh and Benguria, stating that inequalities 1 and 2 hold if the sum at the left-hand side is taken up to \(j=n\), and the numerator at the right-hand side is replaced by \(n\).